1) I feel that it is funny/strange/weird (a feeling about something external)
2) I feel a strong attraction (something external is making you feel something)
3) This morning I feel cold (a personal feeling)

1. I feel sad (personal feeling), I would use 悲しく感します。
2.I feel happy (external), I would use 美人を見ると、嬉しいと感します。

Actually in the first case rather than using 感じる you'd be better off with 思う, which doesn't just mean 'to think' but also 'to feel.' Notice how both these kanji have the 'heart radical in them (心)

Eg.

悲しく思う

Actually this does not mean 'I feel sad' but 'I feel sad about X'. There is a subject to this sentence that isn't clear here.
If you just want to say 'I feel sad' you don't need a verb at all. Just say 「悲しい」 !

Now, 感じる means 'I feel' in more of a "sensation" kind of... sense. So yes, you can say 悲しく感じる or 嬉しく感じる。 But if you search them on google you'll see how these phrases are used. 時の流れを悲しく感じる。 What you're feeling is the flow of time, and sad is just describing the way in which you feel it. If you're talking about your state of mind, 悲しい is enough.